The class blog for Math 3010, fall 2014, at the University of Utah

Tag Archives: Euclidean Geometry

I learned the parallel postulate in middle school. The best known equivalent of the postulate is attributed to Scottish mathematician John Playfair, and it says that “in a plane, given a line and a point not on it, at most one line parallel to the given line can be drawn through the point.”

The reason that I have a special impression on this postulate may be probably due to a popular metaphor in my middle school period. That metaphor related the parallel lines with the mutual feelings between girls and boys: when a girl and a boy cannot stay together, or they do not develop a mutual affection, we say that they are like two parallel lines. No matter what the two parallel lines “do”, they cannot have an interaction. Similarly, for the two unlucky people, no matter what they do, they can never fall in love with each other. I have to say this metaphor describes a tragic situation and sometimes I do not feel satisfied with the “tragic” destinies of the two parallel lines. Fortunately, as my mathematical knowledge grows, I do find that in some other branches of geometry, the seemingly unbreakable law in Euclidean geometry no longer holds. Among the new branches are hyperbolic geometry and elliptic geometry, which will be the main topic of this blog.

Before we talk about non-Euclidean geometry, let me have a brief introduction to the differences between non-Euclidean geometry and Euclidean geometry. The fundamental difference between them lies in the parallel postulate. We already stated a widely adopted equivalent of parallel postulate in the beginning of this article. For two thousand years after Euclid’s work was published, many mathematicians either tried to prove this “fifth postulate” (in Euclid’s Element) or tried to show that it’s not necessarily true. Actually, even in Euclid’s own book, this parallel postulate was left unproved; Also, unlike the first four postulates, the fifth postulate — the “tragic” parallel postulate, was not being used to prove his following theorems in the book. A breakthrough in this topic came out in the 18th century. A Russian mathematician, Nikolai Lobachevsky, developed the hyperbolic geometry. His most famous contributions are in two aspects: he convincingly showed that Euclid’s fifth postulate cannot be proved, and he presented hyperbolic geometry to the world.

In the original parallel postulate, we said for any given line R and point P, there is exactly one line through P that does not intersect R; i.e., parallel to R. In hyperbolic geometry there are at least two distinct lines through P which do not intersect R, rendering the parallel postulate invalid. Hyperbolic geometry may be against common sense at first glance, because usually, our recognition about the shape of a space is limited to Euclidean space. However, hyperbolic geometric space does exist, one example is the saddle space with a constant negative Gaussian Curvature. Hyperbolic space is possible in dimensions that are larger than or equal to two. It is curved — the reason why it differs from Euclidean planes — and is characterized by a constant negative curvature. Euclidean spaces are always with zero curvature. To make it more vivid in my own words (which very likely will not be so rigorous), if we observe a small region in the hyperbolic plane, it looks like just a concave plane. And when you draw a triangle in this concave plane, the sum of its inner angles is always less than 180 degrees. This is also a proved theorem in hyperbolic geometry.

In elliptic geometry, we have the following conclusion: “Given a line L and a point p outside L, there exists no line parallel to L passing through p, and all lines in elliptic geometry intersect.” This means we can never find any parallel lines in elliptic geometry. This kind of geometry together with hyperbolic geometry, perfectly form a counter example of the parallel postulate’s assumption “there is one and only one parallel line…”: in elliptic geometry, there is more than one parallel line, and in hyperbolic geometry, there are none. Examples of elliptic geometry are more common in our real life than hyperbolic geometry. One example is the surface of Earth. A line in such a space becomes a great circle (a circle centered at earth’s core). When you draw a line through point P and if P is away from line (great circle) L, the new line you get will be a new great circle, and it will always have two intersections with great circle L, because any two great circles on the surface of sphere will have two intersections.

Here we have three pictures visualizing the relationship between Euclid’s geometry, hyperbolic geometry and elliptic geometry.

Image: Joshuabowman and Pbroks, via Wikimedia Commons.

The establishment of non-Euclidean geometry is the outcome of many generations’ collective endeavors. For example, classical era’s scholar Proclus commented some attempts to prove the postulate, esp. Those attempts tried to deduce it from the previous four postulates; Arab mathematician Ibn al-Haytham in the 10th century, tried to prove the theorem by contradiction; in the Age of Enlightenment Italian mathematician Giordano Vitale and Girolamo Saccheri both contributed new approaches to this problem although they finally failed; Gauss and Nikolai Lobachevsky (we already mentioned him above) also joined the sequence — the latter finally finished this task by establishing a new geometric branch. This mansion was built over such a long time and I am fortunate to feel part of its grandeur and beauty.

So for those suitors who complain their misfortune that their dream lovers and they are like two parallel lines, I think you are too pessimistic. You can imagine yourself being in a elliptic geometric space. Then as long as you try your best, you will always have an intersection with the other line. I am not sure whether this will convince those guys and give them confidence. For me, I am now feeling happy and believe that everything is possible in our real world, just like that everything is possible in mathematics. The story about seemingly very simple parallel lines do make me feel the power and beauty of mathematics.

As we learned about the Euclidean geometry and its five basic axioms in class, some terms like “straight line”, “circle”, and “right angle” kept jumping in my mind. I thought I had a picture of them. for example, a straight line is as straight as the rope with a ball attached and hang in the air, and a right angle is shown like a corner of a rectangular table. However, as a math major student, such a simple cognition of them is not enough, I hope to have some more mathematical concept to express them.

The Cartesian coordinate system

2.1 The invention of Cartesian coordinates

In the 17th century, René Descartes (Latinized name: Cartesius), a well-known mathematician and philosopher to today’s people all around the world, published his work La Géométrie , in which he made a breakthrough. More concretely, Descartes uses two straight lines that are perpendicular to each other as axes x, y, and uses these axesto measure the positions of any points in a plane.

2.2 The rule of representing a point in Cartesian coordinates

One point in Cartesian coordinates has two parameters: one is the x parameter, the other is the y parameter. To measure the x parameter, we need to draw a straight line y’ parallel to the y axis(we will discuss the definition of parallel in Cartesian coordinates later) that through the point, and then set the x parameter of that point as the number of the intersection of y’ and x axis, for its y parameter, draw a line x’ parallel to the x axis through the point and take the number on the intersection of x’ and y-axis as this point’s y parameter.

2.3 To express a straight line in Cartesian coordinates

A straight line in Euclidean geometry is a straight object with negligible width and depth. So, it is an idealization of such objects in Euclidean geometry. However, in Cartesian coordinates, a line has a strict definition, a straight line is the set of points that satisfies a certain equation. And the line equation usually can be written as:

A*x + B*y + C = 0,

The A, B, and C are the coefficients of x, y, and constant. Moreover, the -A/B is the slope of the straight line, -C/B is the y-intercept of this line, which means the intersection of the y-axes and the line.

So, all above is how we express a straight line in Cartesian coordinates.

In Cartesian coordinates, to express a line we only need one point and a direction. Suppose we have two points a=(A, C) and b=(B, D). By doing a subtraction of the two points, we can get a vector (B – A, D – C). We only need this vector to provides a direction, which is (B – A)/(D – C). So this unique straight line can be expressed as

A right angle in the Cartesian system is always equal to the angle between x and y axes, for x, y axes in the Cartesian system is perpendicular to each others.

“That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.”

For the parallel postulate, it is far more easier to be expressed in Cartesian coordinates, suppose we already know a line as

A*x + B*y + C = 0,

And we have a point (x1, y1) out of the line, have the point and a direction of the other line, the slope is – A/B, and the other line can be described as

-A/B*(x – x1) = y – y1;

And it is easy to know these two lines are parallel, because they have same slope and do not share one point. And by the property of Cartesian coordinates, this is the only line that parallel with the first one.

Conclusion

In Euclidean geometry, some concept are hard to imagine or describe, while Cartesian coordinate make it possible and easy to express, such a great combination of geometry and algebra!

Our goal is to get to the Euler line, a line that passes through a triangle’s circumcenter, centroid, and orthocenter. The line is only determined for non-equilateral triangles; the points coincide in the equilateral case. We’ll look at the three points above.

The circumcenter, centroid, and orthocenter are all “centers” of triangle. But what is a center of a triangle? Surely, it’s not a point equidistant to all points on the triangle. Our triangle would be a circle in that case.

The circumcenter of a triangle ABC is the center O of the circle K that triangle ABC is inscribed in.

Circumcenter O of triangle ABC. Image drawn by me.

The circumcenter is actually the intersection of the three perpendicular bisectors of the triangle: FE, IG, and DH. To see this, first suppose that triangle ABC has a circumscribed circle K with center O. Draw radii AO, BO, and CO to each of the triangles vertices. This creates three smaller triangles AOB, BOC, and AOC. In each of these smaller triangles, drop an altitude from O. For example, in triangle AOB, altitude OD would be dropped. This splits AOB into two smaller triangles that are congruent by SAS, Line OD is perpendicular to AB by construction, and AD = DB. Hence OD is indeed a perpendicular bisector of side AB. Repeating this for other sides shows that the center of the circumscribed circle is the intersection of ABC‘s perpendicular bisectors.

Moreover, the intersection of any to perpendicular bisectors is equidistant from each of the triangle’s vertices. The reader can see this by considering triangle AOC. Perpendicular bisector IG splits AOC into triangles that are congruent by SAS. It follows that lengths AO and OC are equal. Repeat for the other sides. We then see that the intersection of the perpendicular bisectors is equidistant from the triangle’s vertices. Thus the perpendicular bisectors of a triangle uniquely determine its circumcenter.

The centroid is the intersection of a triangle’s three medians, lines drawn from a vertex that bisect the opposite side. As said in class, the centroid is the center of mass for a thin, triangular solid with uniformly distributed mass.

Interestingly, the midpoints of the sides of triangle ABC—the ends of the medians—cut the triangle into four congruent triangles. We will prove this in a roundabout way. Let E be the midpoint of AB. Draw a line EF parallel to AC where F intersects BC. Similarly draw FD parallel to AB. By construction, EFDA and EFCD are parallelograms. Then AD = EF = DC, so D is the midpoint of AC. Similarly, F is the midpoint of BC. The reader can see that the triangles are congruent by repeatedly applying SAS.

Our final center is the orthocenter, the intersection of the three altitudes of a triangle. An altitude is a segment drawn from a vertex that is perpendicular to the opposite side. As with the two previous centers, the intersection of the altitudes at a single point isn’t immediately obvious.

Orthocenter O of triangle ABC. Drawn by me.

We show that the altitudes of triangle ABC intersect. Construct triangle DEF with triangle ABC inscribed in it by making sides DF, FE, and DE parallel respectively to BC, AB, and AC. Draw altitude BK where K intersects DF. Since AC is parallel to DE, BK is perpendicular to DE. Moreover, ADBC and BACE are parallelograms, so DB = AC = AE. Hence BK is a perpendicular bisector of DE. We repeat the argument for the other altitudes of triangle ABC. Then the altitudes of ABC intersect because the perpendicular bisectors of DEF intersect.

There are a few other centers of a triangle that are either irrelevant to the Euler line or take too long to construct (i.e. I’m tired of drawing diagrams). The incenter is the center of the circle inscribed within a triangle. The incenter also turns out to be the center of a triangle’s angle bisectors. The Euler line doesn’t pass through the incenter.

The nine-point circle is the circle that passes through the feet of the altitudes (the end that isn’t the vertex) of a triangle.

Nine-point circle of ABC. Image: Maksim, via Wikimedia Commons.

Strangely, the circle also passes through the midpoints of the sides of its triangle. But that’s not all. The circle passes through the Euler points, the midpoints of the segments joining the triangle’s vertices to the triangle’s orthocenter. Thus the nine-point circle does indeed pass through nine special points of a triangle. The center of the nine-point circle lies on the Euler line.

After all this, we still haven’t proved that the circumcenter, centroid, and orthocenter lie on the same line. We won’t prove this. Here’s a video of the proof by Salman Khan: https://www.youtube.com/watch?v=t_EgAi574sM. The proof uses a few facts about the centers we haven’t discussed, but these facts aren’t too hard to show. Refer back to my four congruent triangles picture. Let O, K, and L respectively be the circumcenter, centroid, and orthocenter of triangle ABC. Then Khan proves that triangle DOK is similar to triangle BLK. This implies angles DKO and CKL are equal, which means O, K, and L lie on the same line.

Sources and cool stuff:

H.S.M. Coxeter and Samuel L. Greitzer’s Geometry Revisited

Paul Zeitz’s The Art and Craft of Problem Solving (Chapter 8 is called “Geometry for Americans”)

After years of math classes, the crazy truth is finally coming out. It is all just in our heads. No way! How can that be? There’s an interesting debate in the world of math. Are math principles the creation of humanity, or are they universal truths that humans discovered? There are compelling arguments on both sides of the debate and both sides have several different sub-levels of thought. In this article, I will discuss them both generally.

The realists maintain that mathematical principles would exist even without people. Humans discovered the principles and brought them into practical use and any intelligent human being could also discover the same principles. This argument is supported by the fact that many cultures have discovered mathematical principles independent of one another. Also, mathematical concepts, such as the Fibonacci sequence and some fractals, occur in nature which would suggest that they exist even without people. Some realists, like the Pythagoreans, believe that the world was created by numbers. The realist point of view can lead to an almost supernatural view of mathematics.

The challenge with mathematical realism is that there is no physical domain where math entities exist. We cannot draw a perfect circle or even a line. We can conceptualize these things in our mind and we can prove them in theory; however, we cannot actually manipulate math entities in the physical world. Many math concepts exist only in the context of our understanding about them and conceptualizing them.

Another view is the anti-realists. They maintain that math is the creation of humans in order to make sense of the world. They recognize that math is an amazing, complex system and that it works as modeled by science. However, some argue that scientific principles could be explained without math. One anti-realist, Hartry Field, demonstrated this by explaining Newton Mechanics without referencing numbers or functions. He explained that, in his opinion, math is fictional and is true only in the context of the story in which it is being told.

So, is it all in our heads? A fiction that was created to explain properties in our world? In reality we may never be able to settle the debate and it may not matter. Math works. That is the beauty of it. In his article “The Unreasonable Effectiveness of Mathematics in the Natural Sciences”, Eugene Wigner observes that

the mathematical formulation of the physicist’s often crude experience leads in an uncanny number of cases to an amazingly accurate description of a large class of phenomena. This shows that the mathematical language has more to commend it than being the only language which we can speak; it shows that it is, in a very real sense, the correct language.

Albert Einstein, 1921. Image: Public domain, via Wikimedia Commons.

Perhaps the best thing to commend mathematics as being real, is that it works. Time and time again, it works. Its principles, laws and theorems, applied over and over, in different settings produce accurate results and predictions. Einstein commented in a 1921 address titled Geometry and Experience, “It is mathematics which affords the exact natural sciences a certain measure of security, to which without mathematics they could not attain.” He explored the question of how math, a product of our mind can be so applicable to the concrete world. He asked, “How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirable appropriate to the object of reality?” Einstein answers this question with the statement, “As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.” He looks specifically at the field of geometry and the need humans have to learn about the relationships of real things to one another. Even though the axioms of geometry are based on “free creations of the human mind”, he says, “Solid bodies are related, with respect to their possible dispositions, as are bodies on Euclidean geometry of three dimensions. Then the propositions of Euclid contain affirmations as to the relations of practically-rigid bodies.” The abstract principles, when applied to “real” world situations prove to be accurate. Einstein continues to explain how the theory of relativity rests on the concepts of Euclidian and non-Euclidian geometry. He challenges the mind to conceptualize a universe which is “finite, yet unbounded”. In the end, it is this ability to use conceptualized principles and apply them to our world that makes mathematics work. So yes, mathematics may be all in our head and it may be a huge puzzle created by humanity, but it is effective, useful, and even beautiful.

Euclidean geometry is the geometry that everyone learns and uses throughout Middle School and High School. In general, geometry is the study of figures, such as points, lines and circles in space. Euclidean geometry is specifically any geometry in which all of Euclid’s postulates and axioms hold. Axioms and postulates are the beginning of reasoning, they are simple statements that are believed to be true without proof. Assuming Euclid’s axioms and postulates found in his book Elements, the rest of Euclid’s classical geometry could be deduced. However, Euclid’s fifth postulate, the parallel postulate, was disconcerting because it was lengthy compared to the rest and not necessarily self evident. Many other ancient mathematicians were dissatisfied with Euclid’s fifth postulate. They thought that it was presumptuous and tried to prove it using lesser axioms or replace it altogether with something they thought to be more intuitive. But their proofs always included an assumption equivalent to the parallel postulate, so for centuries the postulate was assumed to be true.

Centuries passed and the postulate remained unproven; however, development to understand Euclid’s postulate continued into the eighteenth century. Perhaps the most well-known equivalent to the parallel postulate is Playfair’s Axiom, which states “through any point in the plane, there is at most one straight line parallel to a given straight line.” Arguably one of the most influential mathematicians, Carl Friedrich Gauss became interested in proving Euclid’s fifth postulate. After attempting to prove the postulate, he instead took Playfair’s Axiom and altered it, creating a completely new postulate. Gauss’ new postulate stated “Through a given point not on a line, there are at least two lines parallel to the given line through the given point.” With this Gauss had unearthed a completely new space that today is called hyperbolic geometry. However, he chose not to publish any of his results, wishing not to get caught up in any political strife. The work was later published by Johann Bolyai and Nikolay Lobachevesky, who both had academic ties to Gauss.

Shortly after this discovery another type of Non-Euclidean geometry was discovered by Gauss’ student Georg Friedrich Bernhard Riemann. Riemann looked at what would happen when you altered Playfiar’s Axiom in the opposite direction than Gauss. Riemann’s alternate postulate is stated as follows, “through a given point not on a line, there exist no lines parallel to the line through the given point.” With this, what is known as elliptical or spherical geometry was discovered.

Spherical geometry. Image: Anders Sandberg via Flickr.

Spherical geometry provides a somewhat simpler model then hyperbolic geometry. Anyone who has ever attempted to wrap a basketball in paper understands that there are some discrepancies between the two surfaces. For instance, triangles behave quite differently than they do in Euclidean geometry. In hyperbolic space, the angles of a triangle add up to less than 180 degrees, and in spherical space, they add up to more than 180 degrees. Spherical geometry also has other inconsistencies with Euclid’s initial assumptions other than the parallel postulate. In Leonard Mlodinow’s book Euclid’s Window, the author describes how Riemann’s space was inconsistent with other of Euclid’s postulates. For instance, Euclid’s second postulate states that “any line segment can be extended indefinitely in either direction.” In spherical space this is not true; the lines, or great circles, have a limit to their space, namely two pi times the radius of the sphere. Mlodinow describes how Riemann saw this postulate as “only necessary to guarantee that the lines had no bounds, which is true of the great circles.” Also, Euclid’s first postulate became less clear, “Given any two points, a line segment can be drawn with those points as its endpoints.” This postulate can be used to easily describe whether a point is between two other points. However, on the globe, choosing two points on the equator such as Ecuador and Indonesia it is difficult to say whether a third point, Kenya, is “between” them. The problem is that there are two ways to connect the points, one passing over North America and another passing over Africa.

For much of our day to day lives Euclidean geometry works great, because on a local scale we appear to live on a flat world. I can go to a soccer field and trust that it will take four 90 degree turns to walk around the perimeter, or that the Pythagorean theorem will work to describe the path between opposite corners. But looking at a larger scale, the surface that we live on is spherical and has different properties than the flat plane. It is interesting to see how Gauss and Riemann, going against the grain of conventional mathematics, led to new and vast fields of undiscovered mathematics. To me, this shows how mathematics is just as much an experimental science as physics or engineering. These new discoveries of mathematical spaces made possible Einstein’s physical description of the space in which we live. Mlodinow closes his section on Gauss and Riemann saying, “though thoroughly remodeled, geometry continued to be the window to understanding our universe.” Even though the properties of these new geometries differ from classic Euclidean geometry and may have more or less practical use, they are just as important. From Euclid up until Gauss, mathematics was largely pragmatic, but the discovery of these new geometries highlights how math can be appreciated for its own sake.